Friday, June 19, 2015

In my previous post I took a crack at understanding Paul Romer's mathiness critique. As far as the post related to Romer (most of it was devoted to Brad DeLong's careless embellishment of the idea), my assessment was that Romer is basically just frustrated that his ideas have not swept away the competition in the field of growth theory. I do not believe that the academics Romer called out are defending indefensible positions for the sake of academic politics. To me, the present situation looks more like a healthy competition between theories of growth emphasizing different mechanisms. This is just what one would expect in a field where the forces at work are complicated and the answers to important questions are hard to come by.

Judging by his reply to my post here, it seems that Romer has a rather low opinion of me. Evidently, I am an "Euler-theorem denier." Admittedly, this is not the worst thing I've ever been called. But in addition to this, I am apparently motivated to deny the truth of this mathematical proposition because doing so signals my commitment to an academic club of serpents that includes the likes of Nobel prize-winning economists Bob Lucas and Ed Prescott. Paul, you flatter me.

I want to take some time here to address the specific charge leveled against me by Romer:

Andolfatto’s brazen mathiness involves a verbal statement about a mathematical model that flies in the face of an impossibility theorem. No model can do what he claims his does. No model can have a competitive equilibrium with price-taking behavior and partially excludable nonrival goods.

Romer's proposition is stated clearly enough. Now all we have to do is check whether it's valid or not. If I can produce a counterexample, then I will have shown Romer's proposition to be invalid. Let me now produce the counterexample.

Consider an economy where people combine raw labor (n) with skill (x) to produce labor services (e) according to the formula e = xn. Interpret n as hours worked over given period of time and assume, for simplicity, that everyone has the same n. On the other hand, people generally differ in their skill level, x. People with greater skills (or skill sets) are represented by larger values of x.

What makes people operating in the same environment more or less skilled than one other? Well, it could be several things. Consider two laborers, one of which is endowed with a physical tool that doubles labor productivity. Then we can write x = 1 for the one laborer and x = 2 for the other. Now consider two entrepreneurs, one of which is possessed with a "mental tool" (an idea, or some general know-how) that doubles labor productivity. Then we could similarly write x = 1 for the one entrepreneur and x = 2 for the other.

Physical and mental tools can differ along an important dimension called "rivalry." Economists call physical objects "rival"(or "subtractable") goods because a physical tool can be in the possession of only one laborer at any given time. The same need not be true for a mental good like an idea. Suppose that "knowing how to perform a task at level x" requires knowledge of a certain recipe. Unlike a physical good, an idea is not subtracted from your mind if you share it with someone else. If I teach you my calculus tool, this in no way diminishes my capacity to use the same calculus tool (or what amounts to the same thing, a perfect replica of my calculus tool). Economists call goods with this property "non-rival" or "non-subtractable" goods.

Alright then, let's proceed to the next step. Assume that the aggregate production function takes the form Y = aE, where a > 0 is a scalar and E represents the aggregate labor input measured in "efficiency units." That is, E equals the sum of e = xn over a population of size N. Let me normalize n = 1 and define X as the sum of x over population N. In this case, E = XN.

Notice that the aggregate production function exhibits constant returns to scale in E. The function is also linear in N. Of course, the function displays increasing returns in X and N together. That is, if we double both X and N, output Y is more than doubled.

The question Romer might ask at this point is "where are the books in this economy?" What he means by this is why can't the smartest person in this economy (the one with the largest x) not publish his knowledge in a recipe book and sell it to the others for a huge gain? This is an excellent question. The answer to it is that knowledge is not always very easily communicated and absorbed by all members of society in this manner. To the extent that knowledge transfer is difficult, the smartest guy in the room has a temporary monopoly over the idea that generates the highest x. Personally, I am surprised that Romer is so offended by this assumption of a missing market. Any university professor knows that distributing the course text book does not, by itself (re: Lucas quote 2009), lead to a growth of his/her students' knowledge base. Some types of knowledge can be sold, but other types of knowledge must be absorbed through effort, not through simple purchase. It is an empirical question as to which type of knowledge transfer mechanism is more or less important.

Next, I want to impose perfect competition. I do this not because I am wedded to the idea that this is how one must model the economy. I do it because Romer claims that it cannot be done.

But wait a minute...how can I assume perfect competition when the smartest person has a monopoly over his/her x? The answer is simple. The people in this economy are competing over the supply of efficiency units of labor e. There is no monopoly over the supply of labor services, e. Under perfect competition, the equilibrium price of an efficiency unit of labor is given by w* = a. Of course, the measured wage per unit of raw labor (w*x) differs across people according to their skill x. The person with the highest x commands the highest price per unit of raw labor.

I should like Paul to note that Euler's theorem does indeed hold in my economy. The entire output is exhausted as payments for labor services (measured in efficiency units). Would people ever devote costly effort to learn something that would give them a higher x in this competitive economy? Sure, why not? Suppose that some raw labor time can be moved away from work and into a learning activity (l). The opportunity cost of this time is the person's foregone wage w*xl. The benefit is the expected value of learning something new (a higher x).

Anyone with an elementary training in economics can plainly see that the model described above has a competitive equilibrium with price-taking behavior and partially excludable non-rival goods. Ergo, Romer is wrong.

I am led to speculate how Romer might object to my counterexample. I suspect he might claim that the x in my model is not really a non-rival good. I get the feeling he might be defining a non-rival good as an idea that can be costlessly disseminated and instantaneously absorbed throughout the population (unless the use of the idea is protected by patent, etc.). If this is the case, then the argument would seem to be one of semantics.

In any case, I have made my point. Romer's proposition above is invalid. One can model a competitive equilibrium with price-taking and partially excludable non-rival goods. This should not be taken as a criticism against Romer's preferred modeling strategy. I'm actually a fan of his research program. My complaint with Romer's mathiness critique is that it ascribes unseen and unknown ulterior motives to a class of economists that find it fruitful to view growth through the lens of competitive equilibrium.

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PS. I see that Nick Rowe has offered a thoughtful response to Romer's "whack-a-mole" comment.

Monday, June 1, 2015

If you're like me, you're probably still trying to wrap your mind around the debate on "mathiness" in the economics profession. I haven't put an inordinate amount of time into the project, but I have made an honest effort trying to understand the nature of Paul Romer's lamentations.

Just what is mathiness, anyway? I'm not sure that a precise and commonly-accepted definition exists. It's not surprising that such a catchy word was soon interpreted in myriad of conflicting ways. This is what happens with words.

To his credit, Romer soon recognized that the word he invented was being interpreted in ways he did not mean. He offers a sort of mea culpa here: Mathiness and Academic Identity. It is definitely a clarification (and I think, a softening) of his initial position. Here are what I take to be its two main points:

So my objection to mathiness is not a critique of the assumptions or structure of the models that others propose. It is a critique of a style that lets economists draw invalid inferences from the assumptions and structure of a model; a style that authors can use to persuade the reader (and themselves) to adopt conclusions that do not follow by the rules of logic; a style that tolerates wishful thinking instead of precise, clearly articulated reasoning. The mathiness that I point to in the Lucas (2009) paper...involves hand-waving and verbal evasion that is the exact opposite of the precision in reasoning and communication exemplified by Debreu/Bourbaki, and I’m for precision and clarity.

I wrote that the economists I criticize for using mathiness are engaged in a campaign of ACADEMIC politics, not one of national politics. Whatever was true in the past, the now fight is over ACADEMIC group identity. For example, one of the things that the people I criticize are campaigning for is a methodological restriction to models with price-taking. For them, price-taking is dogma. To make the case for this restriction, they are not presenting scientific arguments grounded in logic and evidence.

Hmm. He's critical of a style that lets economists draw incorrect inferences from the assumptions of a model. This is in contrast, I suppose, to all the other styles that permit people to draw incorrect inferences from the assumptions of their pet theories. He gives the example of Lucas (2009), to which I will return in a moment. The other charge is that some economists (them, not us of course) are "dogmatic." They campaign for and adopt methodological restrictions, like price-taking behavior, even when the logic and evidence does not permit it.

The two points above are linked to the same phenomenon: the apparent unwillingness of Romer's competitors in the field of growth theory to see the error of their ways and the superiority of his preferred approach. Mathiness is impeding scientific progress in the field of growth theory (and possibly throughout the economics profession as well).

What is the basis for these charges? As an outsider in the field of growth theory, it's hard for me to say (though I do have a paper in the area, which I'll talk about below). I think that Deitz Vollrath has a reasonable take on the issue here: More on Mathiness. Thankfully, we have Brad DeLong (Putting Economic Models in Their Place) doing his best to embellish the critique and apply it more broadly to macroeconomic theory--a subject more familiar to me.

Let's start with DeLong, who states:

He (Romer) seems, to me at least, to be very worried principally about two aspects of modern economic discourse. The first is to take what is true about one restricted class of theories and generalize it, claiming it is true of all theories and of the world as well.

Is there really any economist who behaves in this manner? Do we all not already know that our assumptions are provisional? That we may make one set of assumptions to address some questions and another set of assumptions to address other questions--that we have no grand unifying theory? So who, pray tell do these folks have in mind? Well, Paul Romer's thesis adviser, for one. Here, they quote Lucas (2009) who writes:

Some knowledge can be ‘embodied’ in books, blueprints, machines, and other kinds of physical capital, and we know how to introduce capital into a growth model, but we also know that doing so does not by itself [my italics] provide an engine of sustained growth.

Now which parts of the quote above constitute a violation of scientific inquiry in your mind? Probably none. But I think it was the last part that got Romer's goat because he might have interpreted it to mean that "we know that Romer's models do not embody a mechanism that can provide an engine of sustained growth." I don't know--that's not the way I interpret the passage. Let's suppose Lucas had instead written "...but we also know that doing so does not necessarily provide an engine of sustained growth." Would that have been better? Can we all just be friends now?

I think it may be instructive to read the section from which the quote was lifted:

Is this really something to get so riled up over? Romer pg. 91 charges Lucas with making "untethered verbal claims"--an opinion he is of course free to hold. But I contend that it is just an opinion. Moreover, it's an opinion over the use of English, not math. It would have been far more useful and compelling, I think, if Romer had instead critiqued the model's internal logic and its ability to interpret the data. Isn't this the way science is supposed to progress? (Romer might reply that he's tried, but that his supervisor just won't listen because he's so dogmatic. I'll return to this possibility below.)

In any case, nowhere do I see evidence of DeLong's outlandish claim that Lucas is peddling an hypothesis he asserts to be true of all theories and of the world as well. But DeLong continues as follows:

Thus what Lucas claims must be true about the world as a matter of correct theory--that the big secret to successful economic growth cannot lie in creating and acquiring the kind of knowledge that gets "'embodied' in books, blueprints, machines..."--rests on the barely-examined decision to restrict attention to only a few kinds of models.

My goodness...this Lucas character...who does he think he is? You know, I have an idea. Why don't we actually go read the offending article and see what he's really up to? To whet your appetite, here's the introductory paragraph:

You know, call me crazy, but that opening passage does not sound like a call to arms to me. Lucas starts off with an interesting question. He bows respectfully to existing growth theory (including Romer's brand). He notes that they are based on important features of reality and that they are interesting theories. It's just that to him...[now, I want you to brace yourself and to please forgive the man...he is, after all, just an academic trying to explore alternative interpretations of the way the world works]...you see, to him, the existing theories of growth are not central (i.e., they are missing something that Lucas thinks is more important).

Lucas then proceeds in this highly provocative way: "In this paper I introduce and partially develop a new model of technical change..." From the way DeLong is writing, you'd think that Lucas had instead written "In this paper, I introduce and develop the grand unifying theory of economic growth and development. I await my second Nobel prize."

You can see how much fun I am having with this. Let me continue, along with DeLong, who writes:

The problem comes with the second principal aspect of "mathiness": to claim that one and only one mode of interaction and one and only one mode of individual decision-making is admissible at the foundation level of economic models. Here Romer attacks the assumption that the only allowable interaction is one of price-taking behavior: selling (or buying) as much as one wants at whatever the single fixed price currently offered by the market is. And here I would attack the assumption that individual decision-making is always characterized by rational expectations.

While there are certainly economists who make liberal use of the price-taking assumption in their research, there are equally many who do not. And I am not aware of any economist who would make the claim that the only allowable interaction is one of price-taking behavior. Many economists do feel more strongly about the desirability of imposing "rational expectations." But there is, of course, a vibrant community of those who feel otherwise. And it's not as if people who employ non-rational expectations models are ostracized from the community--unless you call being appointed a central bank president a form ostracism (see, for example, this piece by Jim Bullard).

Why do people have such a bee in their bonnet when it comes to the assumption of price-taking behavior? It's just a pricing protocol (a mechanism that determines prices). Many macro models, especially in the search and matching literature, use bilateral bargaining protocols to determine prices. Monopolistic competition is an assumed market structure together with an assumed pricing protocol. The pricing protocol there is usually quite restrictive: a monopolistic competitor is only permitted to charge a single price for his product. That is, nonlinear pricing schedules (which would increase both profit and social welfare) are assumed away--despite the overwhelming evidence of their use (e.g., retail and wholesale establishments often apply price discounts on large quantities purchased.)

[Aside: Many people are under the impression that monopoly is necessarily inefficient. This is not a valid conclusion. The conclusion follows from an auxiliary assumption that rules out an optimal nonlinear pricing protocol. Walter Oi makes this point in what is surely one of the best titles ever for an economics article: A Disneyland Dilemma: Two-Part Tariffs for a Mickey Mouse Monopoly.]

For Romer, the issue has to do with (I think) of how to reconcile the costly acquisition of nonrivalous (nonexcludable) ideas with perfect competition. DeLong hints at this when he writes:

Thus Paul Romer sees, in growth theory, the current generation of neoclassical economists grind out paper after paper imposing on the world "the restriction of 0 percent excludability of ideas required for [the] Marshallian external increasing returns" necessary for there to even be a price-taking equilibrium.

But DeLong (and Romer) are (I think) wrong on this dimension, at least, on a technical level. It is in fact possible to write down a growth a model where nonrivalous ideas are partially excludable (subject to costly acquisition) in a competitive equilibrium with price-taking behavior. My paper here (with Glenn MacDonald) constitutes one such example.

[Aside: In an extension of my model with Glenn, entrepreneurs are motivated to discover a technological advancement that, conditional on discovery, is assumed to diffuse rapidly among a small but measurable set of firms. This "small" group of firms on the technological frontier is "large" enough to assume price-taking behavior. The technology leaders earn profits (return on their knowledge capital) while laggards attempt to imitate the leaders--an effort that collectively generates the classic S-shaped diffusion dynamic--a phenomenon ignored by most growth models--including Romer's. An alternative and possibly more appealing set up would have been to permit one firm to have a short-lived monopoly right over a technological discovery. I think it's doubtful that any of our main results would have changed under this alternative specification. It's definitely an interesting question to explore and the alternative specification may have implications for questions that we were not interested in pursuing. The point I want to stress is that we did not assume price-taking behavior out of respect for a dogma. It just turned out to be a convenient way to approach things. And I think the approach is still fine, depending on the set of questions the researcher wants to focus on. Telling me that I must behave otherwise in the interest of science seems rather...what's the word I'm looking for here...I think it starts with a D...].

To many people, the assumption of price-taking behavior and rational expectations is thought to imply no useful role for government policy. In fact, nothing could be further from the truth. It is well-known that the equilibria of noncooperative games are generically suboptimal (so that well-designed interventions can be welfare-improving). For that matter, the equilibrium of a monopolistically competitive model can be efficient (and therefore warrant no government intervention). Adopting any one of these assumptions a priori is not, by itself, going to lead to a predetermined policy recommendation. And yet, DeLong seems to suggest that this is indeed the case when he writes:

And I see, in macroeconomics, paper after paper and banker after banker and industrialist after industrialist and technocrat after technocrat and politician after politician claiming that everything that governments might to to speed recovery must be counterproductive, or at least too risky--because that is what is in the case in a very restricted class of rational-expectations models.

DeLong claims to see (without providing examples), in macroeconomics, "paper after paper" claiming that everything governments do is worse than useless. As evidence for this this, he cites two economists writing in the 1930s. Why not take a look at what people are publishing in high-level journals today? Here is just one example, drawn from the "freshwater" based Journal of Monetary Economics for March 2015. Among the papers published in this issue, I see:

I'll let this evidence speak for itself. What of his suggestion that the "austerity" forces are motivated by what is true in a restricted class of rational-expectations models? I'm afraid it's just another preposterous statement on his part. As he notes, recommendations of "austerity" were being made even in the 1930s. But that was well before rational-expectations models even existed. I might add that economists today frequently recommend government interventions that rely on rational expectations (e.g., Michael Woodford's "forward guidance" policy proposal).

Let me conclude now by returning to Romer. I'm still not absolutely sure what the mathiness critique is really all about. I think that Romer might just be frustrated (possibly with some justification) that his ideas haven't swept away his competition. Because his ideas are so firmly rooted in logic and evidence, the only thing that can evidently explain the continued resistance is academic politics shrouded in a cloak of mathiness.

Well, that's one interpretation and, who knows, maybe there's an element of truth to it. But my preferred interpretation is that what we seem to have here is simply a healthy clash of ideas competing in the marketplace for ideas. Let's not get too frustrated when our preferred (and obviously superior) theory does not sweep away the competition.

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Steve Williamson has his own interesting take on the issue here.

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